A random composition of n appears when the points of a random closed set
R⊂[0,1] are used to separate into blocks n
points sampled from the uniform distribution. We study the number of parts
Kn​ of this composition and other related functionals under the assumption
that R=ϕ(S∙​), where (St​,t≥0) is a
subordinator and ϕ:[0,∞]→[0,1] is a diffeomorphism. We derive the
asymptotics of Kn​ when the L\'{e}vy measure of the subordinator is regularly
varying at 0 with positive index. Specializing to the case of exponential
function ϕ(x)=1−e−x, we establish a connection between the asymptotics
of Kn​ and the exponential functional of the subordinator.Comment: Published at http://dx.doi.org/10.1214/009117905000000639 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org